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Power company fuel supply — Problems and solutions
Electric Power Systems Research, 5 (1982) 151 - :[61
151
P o w e r C o m p a n y Fuel S u p p l y -- Problems and Solutions
J. D. MORGAN
University of Missouri-Rolla, Electrical Engineering Department, Rolla, MO 65401 (U.S.A.) R. A. SMITH
University of Illinois, Electrical Engineering Department, Urbana, IL 61801 (U.S.A.) (Received December 17, 1981)
SUMMARY
2. FUEL SUPPLY MANAGEMENT SUBPROBLEMS
A p o w e r c o m p a n y in the business o f supplying electricity to its customers faces numerous problems associated with p o w e r production. One o f its major problems is to supply its p o w e r plants with fuel and to use that fuel as economically as possible. The purpose o f this paper is to discuss the difficulties that a p o w e r c o m p a n y has in supplying itself with fuel and to propose a technique that will produce a more p r u d e n t and economical use o f fuel. The technique utilizes a multistage c o m p u t e r algorithm, which takes into consideration the p o w e r company's complete fuel supply process from coal mine to busbar. It requires a mathematical modeling o f this process and allows for as much modeling flexibility as possible.
The overall problem of managing fuel supplies for p o w e r companies has been identified as consisting of purchasing, transporting, and using. These basic categories can be classified into the following five subproblems: (1) Which fuel supplies are available? (2) Which p o w e r plants can use them? (3) H o w should they be shipped to the plants? (4) When should they be unloaded when they arrive at the plants? (5) When should they be dispatched? Each of these problems is incorporated into a model of the fuel management process and is described below. The availability of fuel supplies is considered b y determining what fuel supplies are available for use and for how long (long- or short-term contracts). Long-term contracts are characterized by a duration of several months or years and usually are limited to a fuel supply for one p o w e r plant. Spot market contracts are instantaneous; that is, the entire quantity of fuel purchased is transported at once. After the salable fuel supplies have been located, a p o w e r c o m p a n y needs to determine where each can be used within its collection o f p o w e r plants needing fuel. This decision is ordinarily based upon some kind o f qualitative analysis of each fuel. For example, in the case of coal, the analysis would include such diverse qualities as the content of Btu, ash, moisture, and sulfur; slagging characteristics; and grindability. Such an analysis might show that a particular fuel is unacceptable for use at all plants or unacceptable for only a subset
1. DEFINITION OF THE F U E L SUPPLY PROBLEM
The fuel supply problem encountered b y a p o w e r c o m p a n y involves the management of purchasing, transporting, and using fuels for p o w e r plants. Two kinds of fuels are available for p o w e r c o m p a n y use: nuclear and fossil. Of these types, nuclear fuel offers less management flexibility. Rigid control of the nuclear fuel cycle b y various government agencies reduces the purchasing, transporting, and using options available for p o w e r c o m p a n y consideration. In contrast, fossil fuels have less rigid controls and thus offer more options for purchasing, transporting, and using. Because o f the contrast in options, this paper will focus attention on fossil fuel management [ 1 - 131. 0378-7796/82/0000-0000/$02.75
© Elsevier Sequoia/Printed in The Netherlands
152 of plants. Thus, the potential exists for each available fuel to be used at one or more plants, and the power c o m p a n y must decide which plants can receive which fuels and the quantity that can be shipped within restrictions imposed by regulating agencies. Naturally, the minimization of the purchase cost is the goal that guides the decision. The decision of what supplies to use generates the subproblem of how to ship the selected fuels. In the case of natural gas and oil, the options are usually limited to pipeline transport. However, for coal the modes of transportation can be river barge, railroad car, or truck. Depending upon points of origination and destination, fuel suppliers may use one or a combination of these modes. Also, several different routes or modes of shipping may exist for the transportation of a fuel between its origination and destination. The power c o m p a n y must choose the methods and routes that will minimize transportation costs and move the fuel between originations and destinations within an acceptable time frame. Fuel supplies that have been shipped encounter additional problems when t h e y arrive at their destinations. Specifically, when should they be unloaded and stored at the power plant? For coal, this problem is especially important in view of the potential delays in unloading that are likely to occur as a result of bad weather or the breakdown of unloading equipment. These events can lead to demurrage costs, which accumulate as shipping units queue at the plant waiting to be unloaded. The power c o m p a n y must determine a way to minimize or eliminate these costs by appropriately managing the unloading schemes. Once the fuels have been unloaded and stored, the last major decision to be made by the power company is when to dispatch the various fuels from each power plant. This decision is critical, because power companies that dispatch their fuel supplies based upon plant production cost curves must establish a dispatch policy that is economical. An example m e t h o d of dispatch is to use fuel supplies on a 'highest-cost-in-first-out' basis; i.e., the fuel supplies t h a t are the most expensive are dispatched first. All of the major subproblems discussed above necessarily impact upon one another.
That is, the outcome of one decision may be in part a function of a previous or even later decision or outcome. For example, the dispatching decision depends upon having the fuel available for use at the power plant. The availability of the fuel will depend, in part, upon the decision of how to transport it. A variation of a day or more in the arrival of a fuel at a power plant may yield different dispatch decision results and, hence, may influence operating costs. Therefore, an optimizing method that ties as m a n y of the decisions together as possible is needed [ 14].
3. LITERATURE REVIEW The problem of coal supply and use by power companies is one of resource (fuel) allocation. This problem is widely studied and well documented in the field of operations research. As far as the application of resource allocation to a power company's fuel supply is concerned, the most significant work is contained in refs. 15 - 19. Among the early works, Hayward e t al. [15] studied the reduction of the purchase and transportation costs of coal for the Duquesne Light Company. They examined a problem involving four power plants that were supplied by five coal mines via barge and rail. The solution for this problem was previously approached by trial and error and later by determining each plant's fuel supplies on a lowest incremental cost-first purchase basis. In this work, the m e t h o d of linear programming was used to solve the problem. Another paper which applied a transportation problem format was published by Anderson [16] in 1961. He developed a specially designed analog c o m p u t e r to solve the coal supply problem. The computer circuit was designed to solve a transportation problem program, and the coal supply problem solved by the c o m p u t e r was modeled in that form. Gillies e t al. [17] solved the coal supply problem for Ontario Hydro. Coal was ordered as a function of various market factors and Ontario Hydro's water resources. The objective of the ordering procedure was to minimize the total cost of the coal supply. This included the purchase cost, the storage cost
153 of coal at the plants, and the cost of possible shortages. The coal was transported by barge on the Great Lakes, and purchases were made once before the Great Lake shipping season started and again after the Great Lake shipping season started. The latter purchase was made to correct deficiencies in the first. The decision of how much coal to buy for a season was based on the results of three simulation models t h a t modeled the coal supply situation for Ontario Hydro. Parkinson and Taylor [18] presented a discussion of the use o f operational research by Great Britain's Central Electricity Generating Board. The fundamental areas of study include power station construction, fuel purchasing and transport, power station operation, transmission and system operation, and organization. The research of interest here concerns the purchasing and transporting of fuel. The technique used for minimizing the costs related to the supply of fuel was linear programming, and the variables and factors considered in the program were transportation model t y p e constraints o f available supply and demand, transportation costs, plant thermal efficiency, thermal content of the coal available, and handling costs. Finally, Morgan e t al. [19] performed some preliminary research in which t h e y identified m a n y of the parameters and problems of interest in coal supply. They theorized about solving these problems, both singly or collectively, with various techniques. They proposed and tested an elementary computational approach to solve some of the problems and discussed the procedure used for the sensitivity analysis of the results of purchasing schedules. Not all the works cited are specifically applicable to the present discussion. The main theme expressed in most of them pertains to the use of the transportation problem format. This format is too restrictive to account for all the options that are needed for an adequate coal supply model. For example: consider a coal purchase constraint that allows a coal shipment to be sent to only one of several locations and that allows no more than a specified m a x i m u m and no less than a specified minimum a m o u n t o f coal t o be selected for the shipment. The problems here are first to determine a means of describing
whether or not the coal shipment has been purchased, then to assure t h a t the solution algorithm does not consign part of the shipment to each of its possible destinations, and finally to select how much coal within the specified m i n i m u m and m a x i m u m limits should be purchased. Constraints of this kind and others of similar complexity are not easily cast into the form of a transportation problem. Another point to consider is that transportation problem formulations account for purchase and transportation costs but do not account for demurrage cost caused by extended queues at the sources or destinations of the product. In the coal supply process, an opportunity exists to fine tune the shipping schedules through appropriate manipulation of the various constraints and objective functions in order to minimize demurrage costs. All of the references overlook the effect of a supply schedule on the eventual use of the product at the destination. In the case of power system coal supply, this raises the question o f how the coal purchased for a power plant impacts its generation level setting with respect to the other plants in the system for a given load profile. The answer to this question requires a model that relates the power system to a given coal supply schedule. Finally, a number of references simulated processes in order to establish better ways of operating. Simulation techniques for resource allocation problems can be cumbersome, because process parameters must be continuously altered to obtain better solutions. Linear-integer programming is more advantageous than simulation because the parameters are entered as extreme limits, and the algorithm optimizes a process performance within those limits without user interaction. Also, if an optimal solution to a resource allocation problem exists, then most linearinteger programming methods are guaranteed to converge to that solution.
4. THE FUEL DISPATCH ALGORITHM A block diagram of the proposed dispatch algorithm is shown in Fig. 1. The method is divided into four blocks, which include: (1}
154
LIII!)NGTERMSYSTEM FORECASTS h. LOADS B. ENVIRONMENTAL POWERIMPORTS EXPURr
FORCED
MID-TERM SYSTEM A~CASTS A. LOADS B. ENVIRONMEN2 AL c. POWERIblPOR[ a EXPORT
SHORT I~RH SYSIEM FORECASTS A. LO~0S B. ENVIRONMENI'AL c, POWERIMI'OR'I &
EXPOR2
OUIAGES~
FUEL CHARACTERISTICS
FUELSUPPLIES FORSALE LONGAND SHORT TERN
:Y~j2~Y S~S]
7--d ZI
t,osc
MIDrEm~
lEVel
~_J
--_
DISPA3CH
REFINED ESTIMATESOF FUEl CHARACTERISTICS
SHORI TER~M DISPAICH
7J
FUEL S~ STOCKPILES AND CHARACTERISTIC
FUEL E~ STOCKPILE SECURITY
DISPATCH ERROR
~UEL DiSPAICH ERROR
t ECONOMIC POWER I
DISPATCH
tERM
--i
ZJ
FUEL
7 511ORT
1
~
SYSIEM GENERAIOR
~ci UAL
i~SIRED ~UEL~A'~E~Sl
REAL TiME 1 pOWER SYSTEM MEASUREMENT DATA Fig. 1. Flowchart o f economic fuel dispatch algorithm.
the economic power dispatch and system power plants, (2) a short-term dispatch, (3) a mid-term dispatch, and (4) a long-term dispatch.
Economic dispatch The economic dispatch program proposed herein responds to the dynamic nature o f power system operations. It calculates appropriate loss formulas and production cost equation constants in response to new network topologies, changes in fuel supply use, and emission control law restrictions. It accepts as input to its operation all necessary real-time power system measurement data and real-time fuel dispatch data. The real-time measurement data include the following information: all power flows, power plant operating data, network topology data, plant i n p u t - o u t p u t curves, weather data at selected points in the system, and emission control monitoring data. The real-time fuel dispatch data include present Btu content, cost/Btu, fuel quality (sulfur, ash, moisture), and amount o f fuel (in Btu) available for use.
The above information is used in the economic dispatch program to give as precise an assignment of the power plant output levels as possible while taking into account the objectives of minimizing fuel costs, transmission losses, and adverse emissions in real time. The economic dispatch program operates in real time in response to the power system load and exercises real-time control over the individual power plants. This control is illustrated in Fig. 1 as the data control line connecting the economic power dispatch and the system generator blocks. The data line is labeled as 'raise and lower pulses', which refer to the power plant control signals of the Energy Management System computer.
Short-term dispatch The next block operation shown in Fig. 1 is the short-term dispatch. The primary function of this block is to supply the economic dispatch program with data concerning the nature of the fuel being used in real time at the power plants and also to supply these same data to the plant personnel.
155 The short-term dispatch accepts as inputs the results of the mid-term dispatch, certain short-term system forecasts including forecasted loads, environment, p o w e r import and export, refined fuel quality estimates, and an error signal of fuel use rate. All of these data are used to calculate the dispatching o f fuel at p o w e r plants for time periods ranging from seconds to hours in advance of real time. The short-term dispatch uses data on fuel supplies that are available or will be available in real time. A list of t h e fuel supplies in stock at each plant for the day is provided b y the midterm dispatch. This list also includes the cost of each supply and its quantity. Additionally, the short-term dispatch expects t o receive from plant personnel a refined estimate of the quality characteristics of the fuel supplies in stock. The quality includes the content of Btus, sulfur, and ash. The fuel supply data are used to investigate the alternative fuel supply dispatch schedules that can be created to meet the forecasted loads, p o w e r import and export schedules, and environmental constraints. The span of time over which the schedules are to be created is optional and can be specified by p o w e r c o m p a n y operating personnel. The investigation of the schedules considers which fuel supplies to assign to the plants at the most appropriate time. The investigative procedure is an application of dynamic programming, which basically divides a given problem into stages or subproblems and then solves the subproblems sequentially. Selected sequential solutions are combined to form the solution to the original problem [ 2 0 ] . The short-term dispatch algorithm has the additional feature o f being self~orrecting. The self-correction procedure is created from a fuel rate error signal. This quantity is calculated by fuel use rate data gathered in real time from the system p o w e r plants. The actual fuel use rate data are subtracted from the desired fuel use rate, which is calculated by dividing the short-term fuel dispatch schedules b y their given time intervals. They are then transmitted to the short-term dispatch. The short-term dispatch program monitors this error signal and, if it becomes too large in magnitude, the program recalculates the short-term dispatch schedule and updates the economic power dispatch pro-
gram and the system p o w e r plants with the new data. These calculations and errors are saved b y the short-term dispatch and used in subsequent calculations in which the quantities of fuels in stock are reevaluated. The fuel rate error signal is generated as a result of inaccuracies in the p o w e r system's forecasted data. As weather patterns and load patterns change, the real-time economic dispatch program assigns generation levels that do not correspond with those predicted by the short-term dispatch. Hence, fuel is used more or less than predicted.
Mid-term dispatch The next portion of Fig. 1 is the mid-term dispatch function block. This block supplies the coarse daily fuel dispatch to the shortterm dispatch at the end of the day before real-time operation of short-term dispatch. It calculates the fuel dispatch for time periods o f from several hours to several days. The operation o f the mid-term dispatch, therefore, is fundamentally similar to the short-term dispatch. It is given a time period of interest and contains additional fuel supply and p o w e r system data. The fuel Supply data include a p o w e r plant fuel allocation schedule as calculated b y the long-term dispatch program these data show quantities, destinations, and arrival dates of recently made fuel purchases -- refined estimates of fuel quality characteristics including Btus, ash, and sulfur content, and a fuel dispatch error. The p o w e r system data include mid-term system forecasts, which contain information on load environment, and power import/export. The function of the mid-term dispatch is to create a list o f available fuel supplies at each p o w e r plant that may be used over an assigned time interval of interest. The calculation procedure is a linear-integer programming approach, which lists all feasible fuel supply schedules. Each schedule is then tested over the time interval of interest for the forecasted system conditions. The time interval is subdivided into segments, and system operating conditions are assumed constant during the interval. Economic dispatch programs are executed at each interval. The values of the objective functions of the economic dispatch program are summed to yield a total cost of operation for each
156 feasible schedule. The schedule with the minimum cost is the one sent to the shortterm dispatch. The error signal generated and used as input to the mid-term dispatch is the difference between each day's desired fuel dispatch and each day's actual fuel dispatch. This quantity is used to adjust the recorded quantities of fuel in stock at each plant in order that the mid-term dispatch can create a more accurate estimate of fuel stockpile quantities.
Long-term dispatch The long-term dispatch algorithm is a set of computer programs developed for the purpose o f selecting from a list of available fuel purchases a combination o f purchases that minimizes fuel supply process costs subject to fuel supply process constraints. Three c o m p u t e r programs are used in the minimization process, one for each of the three parts into which the algorithm is conveniently divided. These programs are: (1) a modified zero-one integer program, (2) a dual all-integer program, and (3) an economic dispatch program. The execution sequence for these programs is shown in the block diagram of Fig. 2. The modified z e r o - o n e integer program is executed first, and its results are the inputs to the dual all-integer program. The dual allinteger program is executed second, and its results are the inputs to the economic dispatch program. Finally, the economic dispatch program is executed, and its results when coupled with the results of the dual allinteger program yield the m i n i m u m cost solution. Each o f these programs performs a distinct task that is an integral part of the fuel supply optimization process. The modified zero-one integer program is used to list all feasible shipping schedules as constrained by the available fuel purchases, possible destinations of each fuel purchase, Modified Zero-One Integer Program
DuolAIIInteger Progra m
Fig. 2. E x e c u t i o n s e q u e n c e o f t h e a l g o r i t h m .
Economic Dispatch Progra m
and power plant fuel demands. A feasible shipping schedule is a group of fuel purchases selected to supply fuel to power plants such that the power plant fuel demands are met. In general, more than one feasible shipping schedule exists for a given set of fuel purchases and power plant demands. This situation occurs because of two characteristics of the fuel supply problem: (1) the power plant fuel demands may be less than the fuel purchase supply such that several combinations of fuel purchase selections, i.e., feasible shipping schedules, may satisfy plant fuel demands; and (2) each fuel purchase may have more than one power plant destination, which may also serve to increase the number of feasible shipping schedules through interaction with reason (1). The execution of the dual all-integer program follows that of the modified zero-one integer program. It requires as inputs a linear objective function and linear constraint set both of which have integer coefficients on every variable. In general, these equations have the following form: min. or max. f =
Cl X 1 + C 2 X 2 + ... + CrX r
subject to: ail Xl
+ a i 2 x 2 + • .. + a i r X r
~
bi,
= bi
o r ~>
bi
i = 1, ..., m
bi, ci, a U, Xi integers The dual all-integer program is executed for each feasible shipping schedule found by the modified z e r o - o n e integer program. The results show the a m o u n t of fuel purchased in each shipment that will minimize purchase and shipping costs and satisfy limits, and the fuel unloading order at each plant that will minimize demurrage costs subject to each plant's unloading capacities. The execution of the economic dispatch program follows that of the dual all-integer program. The purpose of the economic dis-
Solution
157 TABLE 1 Coal p u r c h a s e d a t a : p a r t 1 Pur. No.
Opt.
Dest. plant
Ship. No.
$ FOB cost/ton
Mode*
U n i t cap. (ton)
Min. (Am. ton)
Max. (Am. t o n )
A B C D E F G
1 1 1 1 1 1 2
H
2
I
3
J
3
K
4
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
1 2 3 4 5 6 7 8 9 10 11 12 13 13 13 14 14 14 15 16 16
25 28 27 28 27 30 31 33 31 25 23 25 26 28 26 27 25 27 28 30 28
R B R B B R B B R R B R R B R R B R T B R
100 1000 50 1000 1000 50 1000 1000 50 100 1000 50 100 1000 50 100 1000 50 10 1000 50
10000 12000 5000 5000 5000 2000 6000 6000 6000 4000 4000 4000 1000 1000 1000 2000 2000 2000 1000 1000 1000
10000 12000 5000 10000 15000 4000 10000 10000 10000 6000 6000 6000 12000 12000 12000 8000 8000 8000 2000 2000 2000
* R = railroad ; B = b a r g e ; T = t r u c k .
TABLE 2 Coal p u r c h a s e d a t a : p a r t 2 Pur. No.
A B C D E F G
H
I
J
K
Ship. N o .
1 2 3 4 5 6 7 8 9 10 11 12 13 13 13 14 14 14 15 16 16
Arr. d a t e
1 2 1 2 1 2 4 5 4 2 2 4 4 3 3 4 3 3 1 6 6
D e m u r r a g e cost p e r u n i t p e r d a y 1
2
3
4
5
6
7
0
2 0 4 0 30 0
2 10 4 30 30 3
4 10 4 30 40 3 0
0 0
1 10
4 20 4 40 40 8 60 0 6 5 30 7 9 50 3 5 20 7 4
6 20 4 40 50 9 60 6O 6 7 50 7 9 90 3 5 70 8 4 0 0
6 30 4 50 60 10 60 60 6 9 70 7 9 130 3 6 70 9 4 60 5
0 0
0 0
0
4
0 0 4
0 3 20 0 0 10 3 0 20 6 4
158 TABLE 3 Feasible purchase schedule Feas. Soln. No.
Shipment No. list
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5 5 5 5 5 5 5
p a t c h p r o g r a m is t o d e t e r m i n e t h e p o w e r system's operating costs for each purchasing schedule over a known time period. The procedure for the execution of the d u a l a l l - i n t e g e r p r o g r a m , w h i c h is f o l l o w e d by the execution of the economic dispatch p r o g r a m , is u s e d f o r e a c h f e a s i b l e p u r c h a s i n g s c h e d u l e f o u n d in t h e m o d i f i e d z e r o - o n e i n t e g e r p r o g r a m . T h e r e s u l t s o f t h e d u a l allinteger p r o g r a m yield purchase, t r a n s p o r t , and demurrage costs, and the results of the economic dispatch yield total energy production costs for each feasible purchasing s c h e d u l e . T h e t o t a l e n e r g y p r o d u c t i o n c o s t is found by summing the purchase, transport, d e m u r r a g e , and p o w e r p r o d u c t i o n costs. The o p t i m u m p u r c h a s e s c h e d u l e is t h e s c h e d u l e t h a t has t h e s m a l l e s t s u m o f c o s t s [ 2 0 - 2 2 ] . F o r purposes o f illustration, a small p r o b l e m f o r l o n g - t e r m d i s p a t c h is given b e l o w . T h i s p r o b l e m e x e r c i s e s as m a n y s h i p p i n g o p t i o n s as p o s s i b l e a n d c o n s i s t s o f three power plants interconnected into a five-bus p o w e r s y s t e m n e t w o r k a n d serviced b y eleven available purchases o f coal for a t i m e s p a n o f seven d a y s . T a b l e s 1 - 3 o f ref. 23 a n d T a b l e s 1 a n d 2 o f t h i s p a p e r s h o w t h e p e r t i n e n t system and coal supply data. The execution of the modified zero-one i n t e g e r p r o g r a m s h o w s 14 p u r c h a s e s c h e d u l e s ( T a b l e 3) t h a t c a n m e e t t h e p o w e r p l a n t ' s c o a l requirements. E a c h p u r c h a s e s c h e d u l e is t r e a t e d as a s e p a r a t e p r o b l e m f r o m t h i s p o i n t o n in t h e a l g o r i t h m . The e x e c u t i o n of the dual all-integer program follows n e x t for each feasible purchase
6 6 6 6 6 6 6 6 6 6 6 6 6 6
7 7 7 7 13 13 13 13 13 13 13 13 13 13
15 15 16 16 8 8 8 8 9 9 9 9 9 9
14 14 14 14 10 10 15 12 10 10 15 15 12 16
11 12 12 11 15 16 12 16 15 16 12 11 16 11
13 13 13 13 14 14 14 14 14 14 14 14 14 14
s c h e d u l e . This stage o f e x e c u t i o n is u s e d t o minimize the purchase, transport, and demurrage c o s t s f o r e a c h s c h e d u l e . T h e s u m o f t h e s e q u a n t i t i e s is r e f e r r e d t o as p r o c u r e m e n t c o s t s . A t y p i c a l s o l u t i o n is s h o w n in T a b l e s 4 a n d 5. Table 4 shows the amount of coal purchased from each coal supply, and Table 5 shows the unloading schedule for the chosen supplies a l o n g w i t h a n y i n c u r r e d d e m u r r a g e costs. TABLE 4 Amount of coal ordered from each coal purchase for feasible solution No. 1 Shipment No.
Tons ordered
1 2 3 4 5 6 7 8 9 10 11 12 13 13 13 14 14 14 15 16 16
10000 12000 5000 10000 15000 4000 9000 0 0 0 6000 0 0 0 12000 0 5000 3000 2000 0 0
to to to to to to
power power power power power power
plant plant plant plant plant plant
1 2 3 1 2 3
to power plant 2 to power plant 3
Total cost
$ 2 517 000
159
The types of data shown in Tables 4 and 5 are used to generate the fuel cost curves for the economic dispatch program. The economic dispatch program is executed once for each feasible solution. The results of these executions are the system operating costs. These costs are listed with the procurement costs in Table 6. Examination of the Table shows that feasible solution number 14 has the lowest total cost. Therefore, this solution is the optimal purchase scheme. The purchasing and unloading schemes created for solution 14 are shown in Tables 7 and 8.
5. CONCLUSIONS
Fuel supply management problems can be mathematically modeled for operations research type solution techniques. The model is capable of enumerating management options for purchasing, transporting, and using fuel and of selecting the best option for
implementation. The resulting selection allows control of the fuel supply from its sources to p o w e r plant busbars. TABLE 6 Total costs of all feasible purchase schedules Feas. soln.
Procurement cost (kS*)
Operating cost (kS*)
Total cost (kS*)
1 2 3 4 5 6 7 8 9 10 11 12 13 14
2518 2530 2529 2517 2554 2553 2554 2553 2530 2529 2531 2518 2530 2517
3532 3547 3548 3531 3572 3570 3566 3570 3537 3531 3540 3522 3537 3518
6051 6077 6077 6048 6126 6123 6120 6123 6067 6060 6070 6040 6067 6035
*Rounded to nearest integer.
TABLE 5 Unloading schedule for feasible solution No. 1 Power plant 1 Ship. No.
No. o f shipping units unloaded per day Day 1
1 4 7 15
80
40
Day 2
Day 3
20 7
3
40
40
Day 4
7 40
Demurrage cost ($) Day 5
Day 6
Day 7
2 40
960
Power plant 2 Ship. No.
No. o f shipping units unloaded per day Day 1
2 5 14 11
10
Day 2 5 5
Day 3
Day 4
Demurrage cost ($) Day 5
Day 6
Day 7
7
3
5 3
30
Power plant 3 Ship. No.
No. o f shipping units unloaded per day Day 1
3 6 14 13
Day 2
Day 3
Day 4
Demurrage cost ($) Day 5
Day 6
100
100
Day 7
100 80 60 40
300
160 TABLE 7 A m o u n t o f coal ordered from each coal purchase for feasible solution No. 14 Shipment No.
Tons ordered
1 2 3 4 5 6 7 8 9 10 11 12 13 13 13 14 14 14 15 16 16
10000 12000 5000 10000 15000 4000 0 0 9000 0 6000 0 11000 0 1000 0 5000 3000 0 0 2000
to to to to to to
power power power power power power
plant plant plant plant plant plant
1 2 3 1 2 3
to power plant 2 to power plant 3
Total cost
REFERENCES I. Elgerd, Electric Energy Systems Theory: An Introduction, McGraw-Hill, New York, 1965,
10.
564 pp. 2 W. Mayer, Nuclear energy -- availability, reliability, and safety, Electrical Board o f Trade
Energy Leadership Seminar, St. Louis, Missouri, February 14, 1974, 22 pp. 3 Committee on U.S. Energy Outlook. Other Energy Resources Subcommittee. Coal Task Group, U.S. Energy Outlook Coal Availability, National Petroleum Council, Washington, 1973, 287 pp. 4 G. D. Friedlander, A comeback for Reddy kilowatt? IEEE Spectrum, 9 (1972) 44 - 50. 5 There's a scrubber in your future, Electr. WorM, 181 (1) (1974) 9. 6 Higher fuel prices, taxes expected under Simon, Electr. WorM, 181 (1) (1974) 25 - 26. 7 EPA pushes coal use to save oil, Electr. World, 181 (1) (1973) 26 - 27. 8 Energy crisis alters power use pattern, Electr. World, 181 (1) (1974) 65. 9 R. Wilson and W. Jones, Energy, Ecology and the Environment, Academic Press, New York, 1974, 353 pp. 10 Environment, Electr. WorM, 181 (8) {1974) 25. 11 Hot debate over basics, Time, 16 (3) (1975) 4 2 - 43. 12 M. O. Young, The promise and problems of coal,
$ 2 517 000
TABLE 8 Unloading schedule for feasible solution No. 14 Power plant 1 Ship. No.
Day 1 1 4 13
Demurrage cost ($)
No. of shipping units unloaded per day
80
Day 2 20 7
Day 3
Day 4
Day 5
Day 6
Day 7
3 80
30
Power plant 2 Ship. No.
No. o f shipping units unloaded per day Day I
2 5 11 14
10
Day 2 5 5
Day 3
Day 4
Demurrage cost ($) Day 5
Day 6
Day 7
7 3
3 5
30
Power plant 3 Ship. No.
No. o f shipping units unloaded per day Day 1
3 6 13 9 16 14
Day 2
Day 3
Demurrage cost ($)
Day 4
Day 5
100
80
Day 6
100 80 20 40 60
Day 7
161
13 14
15
16
17
Electrical Board o f Trade -- National Electric Week Program, St. Louis, Missouri, February 14, 1974, 16 pp. 'Coal by wire' dwindles, conversions are slow, Eleetr. World, 181 (7) (1974) 25 - 26. G. C. Ferrel, Coal, Economics and the Environment: Tradeoffs in the Coal Electric Cycle, International Institute for Applied Systems Analysis, Salzburg, Austria, 1978. A. P. Hayward, C. E. Taylor, R. H. Kerr and L. K. Kirchmayer, Minimization of fuel costs by the technique of linear programming, Trans. Am. Inst. Electr. Eng., Part 3, 76 (1958) 1288 - 1295. E. G. Anderson, An electronic analogue computer for a coal transportation problem, Proc. Inst. Electr. Eng., Part B, 108 (1961) 43 - 47. D. K. A. Gillies, J. G. C. Ternpleton and W. H. Winter, Optimum safety stock in a coal-ordering problem, Can. Oper. Res. Soc., 3 (1965) 2 9 - 45.
18 W. L. Parkinson and D. Taylor, Operational research in the Central Electricity Generating Board, Oper. Res. Q., 16 (1965) 133 - 143. 19 J. D. Morgan, R. A. Smith and L. S. VanSlyck, Coal supply optimization on the A E P system, Proceedings o f the Midwest Power Symposium, October 21 - 22, 1974. 20 B. E. Gillett, Introduction to Operations Research: A Computer-Oriented Algorithmic Approach, McGraw-Hill, New York, 1976, 617 pp. 21 R. A. Smith, Optimal fuel supply control for electrical energy systems -- a real-time computer technique, Ph.D. Dissertation, University of Missouri-Rolla, 1976, 140 pp. 22 N. Hokierti, A master data bank in power systems, 1975. 23 J. D. Morgan and R. A. Smith, Power company fuel supply -- sensitivity analysis, Electr. Power Syst. Res., 5 (1982) 1 - 11.
151
P o w e r C o m p a n y Fuel S u p p l y -- Problems and Solutions
J. D. MORGAN
University of Missouri-Rolla, Electrical Engineering Department, Rolla, MO 65401 (U.S.A.) R. A. SMITH
University of Illinois, Electrical Engineering Department, Urbana, IL 61801 (U.S.A.) (Received December 17, 1981)
SUMMARY
2. FUEL SUPPLY MANAGEMENT SUBPROBLEMS
A p o w e r c o m p a n y in the business o f supplying electricity to its customers faces numerous problems associated with p o w e r production. One o f its major problems is to supply its p o w e r plants with fuel and to use that fuel as economically as possible. The purpose o f this paper is to discuss the difficulties that a p o w e r c o m p a n y has in supplying itself with fuel and to propose a technique that will produce a more p r u d e n t and economical use o f fuel. The technique utilizes a multistage c o m p u t e r algorithm, which takes into consideration the p o w e r company's complete fuel supply process from coal mine to busbar. It requires a mathematical modeling o f this process and allows for as much modeling flexibility as possible.
The overall problem of managing fuel supplies for p o w e r companies has been identified as consisting of purchasing, transporting, and using. These basic categories can be classified into the following five subproblems: (1) Which fuel supplies are available? (2) Which p o w e r plants can use them? (3) H o w should they be shipped to the plants? (4) When should they be unloaded when they arrive at the plants? (5) When should they be dispatched? Each of these problems is incorporated into a model of the fuel management process and is described below. The availability of fuel supplies is considered b y determining what fuel supplies are available for use and for how long (long- or short-term contracts). Long-term contracts are characterized by a duration of several months or years and usually are limited to a fuel supply for one p o w e r plant. Spot market contracts are instantaneous; that is, the entire quantity of fuel purchased is transported at once. After the salable fuel supplies have been located, a p o w e r c o m p a n y needs to determine where each can be used within its collection o f p o w e r plants needing fuel. This decision is ordinarily based upon some kind o f qualitative analysis of each fuel. For example, in the case of coal, the analysis would include such diverse qualities as the content of Btu, ash, moisture, and sulfur; slagging characteristics; and grindability. Such an analysis might show that a particular fuel is unacceptable for use at all plants or unacceptable for only a subset
1. DEFINITION OF THE F U E L SUPPLY PROBLEM
The fuel supply problem encountered b y a p o w e r c o m p a n y involves the management of purchasing, transporting, and using fuels for p o w e r plants. Two kinds of fuels are available for p o w e r c o m p a n y use: nuclear and fossil. Of these types, nuclear fuel offers less management flexibility. Rigid control of the nuclear fuel cycle b y various government agencies reduces the purchasing, transporting, and using options available for p o w e r c o m p a n y consideration. In contrast, fossil fuels have less rigid controls and thus offer more options for purchasing, transporting, and using. Because o f the contrast in options, this paper will focus attention on fossil fuel management [ 1 - 131. 0378-7796/82/0000-0000/$02.75
© Elsevier Sequoia/Printed in The Netherlands
152 of plants. Thus, the potential exists for each available fuel to be used at one or more plants, and the power c o m p a n y must decide which plants can receive which fuels and the quantity that can be shipped within restrictions imposed by regulating agencies. Naturally, the minimization of the purchase cost is the goal that guides the decision. The decision of what supplies to use generates the subproblem of how to ship the selected fuels. In the case of natural gas and oil, the options are usually limited to pipeline transport. However, for coal the modes of transportation can be river barge, railroad car, or truck. Depending upon points of origination and destination, fuel suppliers may use one or a combination of these modes. Also, several different routes or modes of shipping may exist for the transportation of a fuel between its origination and destination. The power c o m p a n y must choose the methods and routes that will minimize transportation costs and move the fuel between originations and destinations within an acceptable time frame. Fuel supplies that have been shipped encounter additional problems when t h e y arrive at their destinations. Specifically, when should they be unloaded and stored at the power plant? For coal, this problem is especially important in view of the potential delays in unloading that are likely to occur as a result of bad weather or the breakdown of unloading equipment. These events can lead to demurrage costs, which accumulate as shipping units queue at the plant waiting to be unloaded. The power c o m p a n y must determine a way to minimize or eliminate these costs by appropriately managing the unloading schemes. Once the fuels have been unloaded and stored, the last major decision to be made by the power company is when to dispatch the various fuels from each power plant. This decision is critical, because power companies that dispatch their fuel supplies based upon plant production cost curves must establish a dispatch policy that is economical. An example m e t h o d of dispatch is to use fuel supplies on a 'highest-cost-in-first-out' basis; i.e., the fuel supplies t h a t are the most expensive are dispatched first. All of the major subproblems discussed above necessarily impact upon one another.
That is, the outcome of one decision may be in part a function of a previous or even later decision or outcome. For example, the dispatching decision depends upon having the fuel available for use at the power plant. The availability of the fuel will depend, in part, upon the decision of how to transport it. A variation of a day or more in the arrival of a fuel at a power plant may yield different dispatch decision results and, hence, may influence operating costs. Therefore, an optimizing method that ties as m a n y of the decisions together as possible is needed [ 14].
3. LITERATURE REVIEW The problem of coal supply and use by power companies is one of resource (fuel) allocation. This problem is widely studied and well documented in the field of operations research. As far as the application of resource allocation to a power company's fuel supply is concerned, the most significant work is contained in refs. 15 - 19. Among the early works, Hayward e t al. [15] studied the reduction of the purchase and transportation costs of coal for the Duquesne Light Company. They examined a problem involving four power plants that were supplied by five coal mines via barge and rail. The solution for this problem was previously approached by trial and error and later by determining each plant's fuel supplies on a lowest incremental cost-first purchase basis. In this work, the m e t h o d of linear programming was used to solve the problem. Another paper which applied a transportation problem format was published by Anderson [16] in 1961. He developed a specially designed analog c o m p u t e r to solve the coal supply problem. The computer circuit was designed to solve a transportation problem program, and the coal supply problem solved by the c o m p u t e r was modeled in that form. Gillies e t al. [17] solved the coal supply problem for Ontario Hydro. Coal was ordered as a function of various market factors and Ontario Hydro's water resources. The objective of the ordering procedure was to minimize the total cost of the coal supply. This included the purchase cost, the storage cost
153 of coal at the plants, and the cost of possible shortages. The coal was transported by barge on the Great Lakes, and purchases were made once before the Great Lake shipping season started and again after the Great Lake shipping season started. The latter purchase was made to correct deficiencies in the first. The decision of how much coal to buy for a season was based on the results of three simulation models t h a t modeled the coal supply situation for Ontario Hydro. Parkinson and Taylor [18] presented a discussion of the use o f operational research by Great Britain's Central Electricity Generating Board. The fundamental areas of study include power station construction, fuel purchasing and transport, power station operation, transmission and system operation, and organization. The research of interest here concerns the purchasing and transporting of fuel. The technique used for minimizing the costs related to the supply of fuel was linear programming, and the variables and factors considered in the program were transportation model t y p e constraints o f available supply and demand, transportation costs, plant thermal efficiency, thermal content of the coal available, and handling costs. Finally, Morgan e t al. [19] performed some preliminary research in which t h e y identified m a n y of the parameters and problems of interest in coal supply. They theorized about solving these problems, both singly or collectively, with various techniques. They proposed and tested an elementary computational approach to solve some of the problems and discussed the procedure used for the sensitivity analysis of the results of purchasing schedules. Not all the works cited are specifically applicable to the present discussion. The main theme expressed in most of them pertains to the use of the transportation problem format. This format is too restrictive to account for all the options that are needed for an adequate coal supply model. For example: consider a coal purchase constraint that allows a coal shipment to be sent to only one of several locations and that allows no more than a specified m a x i m u m and no less than a specified minimum a m o u n t o f coal t o be selected for the shipment. The problems here are first to determine a means of describing
whether or not the coal shipment has been purchased, then to assure t h a t the solution algorithm does not consign part of the shipment to each of its possible destinations, and finally to select how much coal within the specified m i n i m u m and m a x i m u m limits should be purchased. Constraints of this kind and others of similar complexity are not easily cast into the form of a transportation problem. Another point to consider is that transportation problem formulations account for purchase and transportation costs but do not account for demurrage cost caused by extended queues at the sources or destinations of the product. In the coal supply process, an opportunity exists to fine tune the shipping schedules through appropriate manipulation of the various constraints and objective functions in order to minimize demurrage costs. All of the references overlook the effect of a supply schedule on the eventual use of the product at the destination. In the case of power system coal supply, this raises the question o f how the coal purchased for a power plant impacts its generation level setting with respect to the other plants in the system for a given load profile. The answer to this question requires a model that relates the power system to a given coal supply schedule. Finally, a number of references simulated processes in order to establish better ways of operating. Simulation techniques for resource allocation problems can be cumbersome, because process parameters must be continuously altered to obtain better solutions. Linear-integer programming is more advantageous than simulation because the parameters are entered as extreme limits, and the algorithm optimizes a process performance within those limits without user interaction. Also, if an optimal solution to a resource allocation problem exists, then most linearinteger programming methods are guaranteed to converge to that solution.
4. THE FUEL DISPATCH ALGORITHM A block diagram of the proposed dispatch algorithm is shown in Fig. 1. The method is divided into four blocks, which include: (1}
154
LIII!)NGTERMSYSTEM FORECASTS h. LOADS B. ENVIRONMENTAL POWERIMPORTS EXPURr
FORCED
MID-TERM SYSTEM A~CASTS A. LOADS B. ENVIRONMEN2 AL c. POWERIblPOR[ a EXPORT
SHORT I~RH SYSIEM FORECASTS A. LO~0S B. ENVIRONMENI'AL c, POWERIMI'OR'I &
EXPOR2
OUIAGES~
FUEL CHARACTERISTICS
FUELSUPPLIES FORSALE LONGAND SHORT TERN
:Y~j2~Y S~S]
7--d ZI
t,osc
MIDrEm~
lEVel
~_J
--_
DISPA3CH
REFINED ESTIMATESOF FUEl CHARACTERISTICS
SHORI TER~M DISPAICH
7J
FUEL S~ STOCKPILES AND CHARACTERISTIC
FUEL E~ STOCKPILE SECURITY
DISPATCH ERROR
~UEL DiSPAICH ERROR
t ECONOMIC POWER I
DISPATCH
tERM
--i
ZJ
FUEL
7 511ORT
1
~
SYSIEM GENERAIOR
~ci UAL
i~SIRED ~UEL~A'~E~Sl
REAL TiME 1 pOWER SYSTEM MEASUREMENT DATA Fig. 1. Flowchart o f economic fuel dispatch algorithm.
the economic power dispatch and system power plants, (2) a short-term dispatch, (3) a mid-term dispatch, and (4) a long-term dispatch.
Economic dispatch The economic dispatch program proposed herein responds to the dynamic nature o f power system operations. It calculates appropriate loss formulas and production cost equation constants in response to new network topologies, changes in fuel supply use, and emission control law restrictions. It accepts as input to its operation all necessary real-time power system measurement data and real-time fuel dispatch data. The real-time measurement data include the following information: all power flows, power plant operating data, network topology data, plant i n p u t - o u t p u t curves, weather data at selected points in the system, and emission control monitoring data. The real-time fuel dispatch data include present Btu content, cost/Btu, fuel quality (sulfur, ash, moisture), and amount o f fuel (in Btu) available for use.
The above information is used in the economic dispatch program to give as precise an assignment of the power plant output levels as possible while taking into account the objectives of minimizing fuel costs, transmission losses, and adverse emissions in real time. The economic dispatch program operates in real time in response to the power system load and exercises real-time control over the individual power plants. This control is illustrated in Fig. 1 as the data control line connecting the economic power dispatch and the system generator blocks. The data line is labeled as 'raise and lower pulses', which refer to the power plant control signals of the Energy Management System computer.
Short-term dispatch The next block operation shown in Fig. 1 is the short-term dispatch. The primary function of this block is to supply the economic dispatch program with data concerning the nature of the fuel being used in real time at the power plants and also to supply these same data to the plant personnel.
155 The short-term dispatch accepts as inputs the results of the mid-term dispatch, certain short-term system forecasts including forecasted loads, environment, p o w e r import and export, refined fuel quality estimates, and an error signal of fuel use rate. All of these data are used to calculate the dispatching o f fuel at p o w e r plants for time periods ranging from seconds to hours in advance of real time. The short-term dispatch uses data on fuel supplies that are available or will be available in real time. A list of t h e fuel supplies in stock at each plant for the day is provided b y the midterm dispatch. This list also includes the cost of each supply and its quantity. Additionally, the short-term dispatch expects t o receive from plant personnel a refined estimate of the quality characteristics of the fuel supplies in stock. The quality includes the content of Btus, sulfur, and ash. The fuel supply data are used to investigate the alternative fuel supply dispatch schedules that can be created to meet the forecasted loads, p o w e r import and export schedules, and environmental constraints. The span of time over which the schedules are to be created is optional and can be specified by p o w e r c o m p a n y operating personnel. The investigation of the schedules considers which fuel supplies to assign to the plants at the most appropriate time. The investigative procedure is an application of dynamic programming, which basically divides a given problem into stages or subproblems and then solves the subproblems sequentially. Selected sequential solutions are combined to form the solution to the original problem [ 2 0 ] . The short-term dispatch algorithm has the additional feature o f being self~orrecting. The self-correction procedure is created from a fuel rate error signal. This quantity is calculated by fuel use rate data gathered in real time from the system p o w e r plants. The actual fuel use rate data are subtracted from the desired fuel use rate, which is calculated by dividing the short-term fuel dispatch schedules b y their given time intervals. They are then transmitted to the short-term dispatch. The short-term dispatch program monitors this error signal and, if it becomes too large in magnitude, the program recalculates the short-term dispatch schedule and updates the economic power dispatch pro-
gram and the system p o w e r plants with the new data. These calculations and errors are saved b y the short-term dispatch and used in subsequent calculations in which the quantities of fuels in stock are reevaluated. The fuel rate error signal is generated as a result of inaccuracies in the p o w e r system's forecasted data. As weather patterns and load patterns change, the real-time economic dispatch program assigns generation levels that do not correspond with those predicted by the short-term dispatch. Hence, fuel is used more or less than predicted.
Mid-term dispatch The next portion of Fig. 1 is the mid-term dispatch function block. This block supplies the coarse daily fuel dispatch to the shortterm dispatch at the end of the day before real-time operation of short-term dispatch. It calculates the fuel dispatch for time periods o f from several hours to several days. The operation o f the mid-term dispatch, therefore, is fundamentally similar to the short-term dispatch. It is given a time period of interest and contains additional fuel supply and p o w e r system data. The fuel Supply data include a p o w e r plant fuel allocation schedule as calculated b y the long-term dispatch program these data show quantities, destinations, and arrival dates of recently made fuel purchases -- refined estimates of fuel quality characteristics including Btus, ash, and sulfur content, and a fuel dispatch error. The p o w e r system data include mid-term system forecasts, which contain information on load environment, and power import/export. The function of the mid-term dispatch is to create a list o f available fuel supplies at each p o w e r plant that may be used over an assigned time interval of interest. The calculation procedure is a linear-integer programming approach, which lists all feasible fuel supply schedules. Each schedule is then tested over the time interval of interest for the forecasted system conditions. The time interval is subdivided into segments, and system operating conditions are assumed constant during the interval. Economic dispatch programs are executed at each interval. The values of the objective functions of the economic dispatch program are summed to yield a total cost of operation for each
156 feasible schedule. The schedule with the minimum cost is the one sent to the shortterm dispatch. The error signal generated and used as input to the mid-term dispatch is the difference between each day's desired fuel dispatch and each day's actual fuel dispatch. This quantity is used to adjust the recorded quantities of fuel in stock at each plant in order that the mid-term dispatch can create a more accurate estimate of fuel stockpile quantities.
Long-term dispatch The long-term dispatch algorithm is a set of computer programs developed for the purpose o f selecting from a list of available fuel purchases a combination o f purchases that minimizes fuel supply process costs subject to fuel supply process constraints. Three c o m p u t e r programs are used in the minimization process, one for each of the three parts into which the algorithm is conveniently divided. These programs are: (1) a modified zero-one integer program, (2) a dual all-integer program, and (3) an economic dispatch program. The execution sequence for these programs is shown in the block diagram of Fig. 2. The modified z e r o - o n e integer program is executed first, and its results are the inputs to the dual all-integer program. The dual allinteger program is executed second, and its results are the inputs to the economic dispatch program. Finally, the economic dispatch program is executed, and its results when coupled with the results of the dual allinteger program yield the m i n i m u m cost solution. Each o f these programs performs a distinct task that is an integral part of the fuel supply optimization process. The modified zero-one integer program is used to list all feasible shipping schedules as constrained by the available fuel purchases, possible destinations of each fuel purchase, Modified Zero-One Integer Program
DuolAIIInteger Progra m
Fig. 2. E x e c u t i o n s e q u e n c e o f t h e a l g o r i t h m .
Economic Dispatch Progra m
and power plant fuel demands. A feasible shipping schedule is a group of fuel purchases selected to supply fuel to power plants such that the power plant fuel demands are met. In general, more than one feasible shipping schedule exists for a given set of fuel purchases and power plant demands. This situation occurs because of two characteristics of the fuel supply problem: (1) the power plant fuel demands may be less than the fuel purchase supply such that several combinations of fuel purchase selections, i.e., feasible shipping schedules, may satisfy plant fuel demands; and (2) each fuel purchase may have more than one power plant destination, which may also serve to increase the number of feasible shipping schedules through interaction with reason (1). The execution of the dual all-integer program follows that of the modified zero-one integer program. It requires as inputs a linear objective function and linear constraint set both of which have integer coefficients on every variable. In general, these equations have the following form: min. or max. f =
Cl X 1 + C 2 X 2 + ... + CrX r
subject to: ail Xl
+ a i 2 x 2 + • .. + a i r X r
~
bi,
= bi
o r ~>
bi
i = 1, ..., m
bi, ci, a U, Xi integers The dual all-integer program is executed for each feasible shipping schedule found by the modified z e r o - o n e integer program. The results show the a m o u n t of fuel purchased in each shipment that will minimize purchase and shipping costs and satisfy limits, and the fuel unloading order at each plant that will minimize demurrage costs subject to each plant's unloading capacities. The execution of the economic dispatch program follows that of the dual all-integer program. The purpose of the economic dis-
Solution
157 TABLE 1 Coal p u r c h a s e d a t a : p a r t 1 Pur. No.
Opt.
Dest. plant
Ship. No.
$ FOB cost/ton
Mode*
U n i t cap. (ton)
Min. (Am. ton)
Max. (Am. t o n )
A B C D E F G
1 1 1 1 1 1 2
H
2
I
3
J
3
K
4
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
1 2 3 4 5 6 7 8 9 10 11 12 13 13 13 14 14 14 15 16 16
25 28 27 28 27 30 31 33 31 25 23 25 26 28 26 27 25 27 28 30 28
R B R B B R B B R R B R R B R R B R T B R
100 1000 50 1000 1000 50 1000 1000 50 100 1000 50 100 1000 50 100 1000 50 10 1000 50
10000 12000 5000 5000 5000 2000 6000 6000 6000 4000 4000 4000 1000 1000 1000 2000 2000 2000 1000 1000 1000
10000 12000 5000 10000 15000 4000 10000 10000 10000 6000 6000 6000 12000 12000 12000 8000 8000 8000 2000 2000 2000
* R = railroad ; B = b a r g e ; T = t r u c k .
TABLE 2 Coal p u r c h a s e d a t a : p a r t 2 Pur. No.
A B C D E F G
H
I
J
K
Ship. N o .
1 2 3 4 5 6 7 8 9 10 11 12 13 13 13 14 14 14 15 16 16
Arr. d a t e
1 2 1 2 1 2 4 5 4 2 2 4 4 3 3 4 3 3 1 6 6
D e m u r r a g e cost p e r u n i t p e r d a y 1
2
3
4
5
6
7
0
2 0 4 0 30 0
2 10 4 30 30 3
4 10 4 30 40 3 0
0 0
1 10
4 20 4 40 40 8 60 0 6 5 30 7 9 50 3 5 20 7 4
6 20 4 40 50 9 60 6O 6 7 50 7 9 90 3 5 70 8 4 0 0
6 30 4 50 60 10 60 60 6 9 70 7 9 130 3 6 70 9 4 60 5
0 0
0 0
0
4
0 0 4
0 3 20 0 0 10 3 0 20 6 4
158 TABLE 3 Feasible purchase schedule Feas. Soln. No.
Shipment No. list
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5 5 5 5 5 5 5
p a t c h p r o g r a m is t o d e t e r m i n e t h e p o w e r system's operating costs for each purchasing schedule over a known time period. The procedure for the execution of the d u a l a l l - i n t e g e r p r o g r a m , w h i c h is f o l l o w e d by the execution of the economic dispatch p r o g r a m , is u s e d f o r e a c h f e a s i b l e p u r c h a s i n g s c h e d u l e f o u n d in t h e m o d i f i e d z e r o - o n e i n t e g e r p r o g r a m . T h e r e s u l t s o f t h e d u a l allinteger p r o g r a m yield purchase, t r a n s p o r t , and demurrage costs, and the results of the economic dispatch yield total energy production costs for each feasible purchasing s c h e d u l e . T h e t o t a l e n e r g y p r o d u c t i o n c o s t is found by summing the purchase, transport, d e m u r r a g e , and p o w e r p r o d u c t i o n costs. The o p t i m u m p u r c h a s e s c h e d u l e is t h e s c h e d u l e t h a t has t h e s m a l l e s t s u m o f c o s t s [ 2 0 - 2 2 ] . F o r purposes o f illustration, a small p r o b l e m f o r l o n g - t e r m d i s p a t c h is given b e l o w . T h i s p r o b l e m e x e r c i s e s as m a n y s h i p p i n g o p t i o n s as p o s s i b l e a n d c o n s i s t s o f three power plants interconnected into a five-bus p o w e r s y s t e m n e t w o r k a n d serviced b y eleven available purchases o f coal for a t i m e s p a n o f seven d a y s . T a b l e s 1 - 3 o f ref. 23 a n d T a b l e s 1 a n d 2 o f t h i s p a p e r s h o w t h e p e r t i n e n t system and coal supply data. The execution of the modified zero-one i n t e g e r p r o g r a m s h o w s 14 p u r c h a s e s c h e d u l e s ( T a b l e 3) t h a t c a n m e e t t h e p o w e r p l a n t ' s c o a l requirements. E a c h p u r c h a s e s c h e d u l e is t r e a t e d as a s e p a r a t e p r o b l e m f r o m t h i s p o i n t o n in t h e a l g o r i t h m . The e x e c u t i o n of the dual all-integer program follows n e x t for each feasible purchase
6 6 6 6 6 6 6 6 6 6 6 6 6 6
7 7 7 7 13 13 13 13 13 13 13 13 13 13
15 15 16 16 8 8 8 8 9 9 9 9 9 9
14 14 14 14 10 10 15 12 10 10 15 15 12 16
11 12 12 11 15 16 12 16 15 16 12 11 16 11
13 13 13 13 14 14 14 14 14 14 14 14 14 14
s c h e d u l e . This stage o f e x e c u t i o n is u s e d t o minimize the purchase, transport, and demurrage c o s t s f o r e a c h s c h e d u l e . T h e s u m o f t h e s e q u a n t i t i e s is r e f e r r e d t o as p r o c u r e m e n t c o s t s . A t y p i c a l s o l u t i o n is s h o w n in T a b l e s 4 a n d 5. Table 4 shows the amount of coal purchased from each coal supply, and Table 5 shows the unloading schedule for the chosen supplies a l o n g w i t h a n y i n c u r r e d d e m u r r a g e costs. TABLE 4 Amount of coal ordered from each coal purchase for feasible solution No. 1 Shipment No.
Tons ordered
1 2 3 4 5 6 7 8 9 10 11 12 13 13 13 14 14 14 15 16 16
10000 12000 5000 10000 15000 4000 9000 0 0 0 6000 0 0 0 12000 0 5000 3000 2000 0 0
to to to to to to
power power power power power power
plant plant plant plant plant plant
1 2 3 1 2 3
to power plant 2 to power plant 3
Total cost
$ 2 517 000
159
The types of data shown in Tables 4 and 5 are used to generate the fuel cost curves for the economic dispatch program. The economic dispatch program is executed once for each feasible solution. The results of these executions are the system operating costs. These costs are listed with the procurement costs in Table 6. Examination of the Table shows that feasible solution number 14 has the lowest total cost. Therefore, this solution is the optimal purchase scheme. The purchasing and unloading schemes created for solution 14 are shown in Tables 7 and 8.
5. CONCLUSIONS
Fuel supply management problems can be mathematically modeled for operations research type solution techniques. The model is capable of enumerating management options for purchasing, transporting, and using fuel and of selecting the best option for
implementation. The resulting selection allows control of the fuel supply from its sources to p o w e r plant busbars. TABLE 6 Total costs of all feasible purchase schedules Feas. soln.
Procurement cost (kS*)
Operating cost (kS*)
Total cost (kS*)
1 2 3 4 5 6 7 8 9 10 11 12 13 14
2518 2530 2529 2517 2554 2553 2554 2553 2530 2529 2531 2518 2530 2517
3532 3547 3548 3531 3572 3570 3566 3570 3537 3531 3540 3522 3537 3518
6051 6077 6077 6048 6126 6123 6120 6123 6067 6060 6070 6040 6067 6035
*Rounded to nearest integer.
TABLE 5 Unloading schedule for feasible solution No. 1 Power plant 1 Ship. No.
No. o f shipping units unloaded per day Day 1
1 4 7 15
80
40
Day 2
Day 3
20 7
3
40
40
Day 4
7 40
Demurrage cost ($) Day 5
Day 6
Day 7
2 40
960
Power plant 2 Ship. No.
No. o f shipping units unloaded per day Day 1
2 5 14 11
10
Day 2 5 5
Day 3
Day 4
Demurrage cost ($) Day 5
Day 6
Day 7
7
3
5 3
30
Power plant 3 Ship. No.
No. o f shipping units unloaded per day Day 1
3 6 14 13
Day 2
Day 3
Day 4
Demurrage cost ($) Day 5
Day 6
100
100
Day 7
100 80 60 40
300
160 TABLE 7 A m o u n t o f coal ordered from each coal purchase for feasible solution No. 14 Shipment No.
Tons ordered
1 2 3 4 5 6 7 8 9 10 11 12 13 13 13 14 14 14 15 16 16
10000 12000 5000 10000 15000 4000 0 0 9000 0 6000 0 11000 0 1000 0 5000 3000 0 0 2000
to to to to to to
power power power power power power
plant plant plant plant plant plant
1 2 3 1 2 3
to power plant 2 to power plant 3
Total cost
REFERENCES I. Elgerd, Electric Energy Systems Theory: An Introduction, McGraw-Hill, New York, 1965,
10.
564 pp. 2 W. Mayer, Nuclear energy -- availability, reliability, and safety, Electrical Board o f Trade
Energy Leadership Seminar, St. Louis, Missouri, February 14, 1974, 22 pp. 3 Committee on U.S. Energy Outlook. Other Energy Resources Subcommittee. Coal Task Group, U.S. Energy Outlook Coal Availability, National Petroleum Council, Washington, 1973, 287 pp. 4 G. D. Friedlander, A comeback for Reddy kilowatt? IEEE Spectrum, 9 (1972) 44 - 50. 5 There's a scrubber in your future, Electr. WorM, 181 (1) (1974) 9. 6 Higher fuel prices, taxes expected under Simon, Electr. WorM, 181 (1) (1974) 25 - 26. 7 EPA pushes coal use to save oil, Electr. World, 181 (1) (1973) 26 - 27. 8 Energy crisis alters power use pattern, Electr. World, 181 (1) (1974) 65. 9 R. Wilson and W. Jones, Energy, Ecology and the Environment, Academic Press, New York, 1974, 353 pp. 10 Environment, Electr. WorM, 181 (8) {1974) 25. 11 Hot debate over basics, Time, 16 (3) (1975) 4 2 - 43. 12 M. O. Young, The promise and problems of coal,
$ 2 517 000
TABLE 8 Unloading schedule for feasible solution No. 14 Power plant 1 Ship. No.
Day 1 1 4 13
Demurrage cost ($)
No. of shipping units unloaded per day
80
Day 2 20 7
Day 3
Day 4
Day 5
Day 6
Day 7
3 80
30
Power plant 2 Ship. No.
No. o f shipping units unloaded per day Day I
2 5 11 14
10
Day 2 5 5
Day 3
Day 4
Demurrage cost ($) Day 5
Day 6
Day 7
7 3
3 5
30
Power plant 3 Ship. No.
No. o f shipping units unloaded per day Day 1
3 6 13 9 16 14
Day 2
Day 3
Demurrage cost ($)
Day 4
Day 5
100
80
Day 6
100 80 20 40 60
Day 7
161
13 14
15
16
17
Electrical Board o f Trade -- National Electric Week Program, St. Louis, Missouri, February 14, 1974, 16 pp. 'Coal by wire' dwindles, conversions are slow, Eleetr. World, 181 (7) (1974) 25 - 26. G. C. Ferrel, Coal, Economics and the Environment: Tradeoffs in the Coal Electric Cycle, International Institute for Applied Systems Analysis, Salzburg, Austria, 1978. A. P. Hayward, C. E. Taylor, R. H. Kerr and L. K. Kirchmayer, Minimization of fuel costs by the technique of linear programming, Trans. Am. Inst. Electr. Eng., Part 3, 76 (1958) 1288 - 1295. E. G. Anderson, An electronic analogue computer for a coal transportation problem, Proc. Inst. Electr. Eng., Part B, 108 (1961) 43 - 47. D. K. A. Gillies, J. G. C. Ternpleton and W. H. Winter, Optimum safety stock in a coal-ordering problem, Can. Oper. Res. Soc., 3 (1965) 2 9 - 45.
18 W. L. Parkinson and D. Taylor, Operational research in the Central Electricity Generating Board, Oper. Res. Q., 16 (1965) 133 - 143. 19 J. D. Morgan, R. A. Smith and L. S. VanSlyck, Coal supply optimization on the A E P system, Proceedings o f the Midwest Power Symposium, October 21 - 22, 1974. 20 B. E. Gillett, Introduction to Operations Research: A Computer-Oriented Algorithmic Approach, McGraw-Hill, New York, 1976, 617 pp. 21 R. A. Smith, Optimal fuel supply control for electrical energy systems -- a real-time computer technique, Ph.D. Dissertation, University of Missouri-Rolla, 1976, 140 pp. 22 N. Hokierti, A master data bank in power systems, 1975. 23 J. D. Morgan and R. A. Smith, Power company fuel supply -- sensitivity analysis, Electr. Power Syst. Res., 5 (1982) 1 - 11.